In a large saucepan, melt butter over medium heat. Whisk in flour and cook for about 1 minute to form a roux. Gradually add chicken broth, whisking continuously until thickened.

Remove from heat, stir in sour cream, cumin, and a pinch of salt and pepper until well combined.

In a mixing bowl, combine cooked shredded chicken, diced onion, green chiles, and cilantro. Add 1 cup of the white sauce to the chicken mixture, stirring gently.

Preheat oven to 350°F (175°C). Spread a thin layer of white sauce in a greased baking dish. Fill tortillas with chicken mixture, rolling tightly and placing seam-side down.

Pour remaining white sauce over the top of the tortillas. Sprinkle Monterey Jack and cheddar cheese over the sauce.

Bake uncovered for 25-30 minutes, until bubbly and cheese is golden brown.

Allow to rest for about 5 minutes before serving. Serve warm with additional toppings if desired.